Comparison of matrix-effect corrections for ordinary and uncertainty weighted linear regressions and determination of major element mean concentrations and total uncertainties of sixty-two international geochemical reference materials from wavelength-dispersive X-ray fluorescence spectrometry

Highlights

• Matrix-effect correction equations developed and used for the UWLR model.

• A large number (62) of calibrators used for the XRF calibration.

• A large number (26) of alpha-corrections used in geochemistry.

• Total 99% uncertainties in complex matrices, a novel approach in geochemistry.

Abstract

In X-ray fluorescence (XRF) spectrometry, matrix-effect corrections for two types of linear regressions (ordinary and uncertainty-based weighted) have been performed for the first time and compared for the determination of major elements in rocks and minerals. The analytical data from different laboratories for 62 international geochemical reference materials (GRMs) were first processed to obtain better estimates of the mean values, along with lower uncertainties (narrower confidence limits of the mean), which were used in the XRF calibrations. The blank intensities were subtracted from the response for each GRM and the respective uncertainties were estimated. The uncertainty-based weighted least-squares linear regression (UWLR) model, being statistically more appropriate than the ordinary least-squares linear regression (OLR) model, provided more reliable regression equations than the OLR. All calculations and matrix-effect corrections were achieved through a newly developed online computer program. Similarly, matrix-effect corrections involving 11 α and 26 α (alphas) are presented and compared. The best UWLR method involving 26 α was successfully applied to the 62 GRMs and 4 similarly complex rock matrices as the calibrators. Both evaluations confirmed the usefulness of the UWLR model. This UWLR model was also compared with the OLR 26 α model for 62 GRMs treated as unknowns and shown to perform better. Thus, the UWLR model, along with the proposed matrix-effect correction method of 26 α, can be recommended as the most appropriate procedure for the calibration of XRF instruments, instead of the commonly used OLR models. This is the first time when 26 α are advantageously in a better estimate of influence coefficients used for major element determinations in rock and mineral samples. Because the UWLR calibration is based on a large number of GRMs of many rock and mineral types, it should be useful for all kinds of geological materials.

Introduction

Modern analytical techniques are based on the calibration of instrumental response as a function of elemental concentrations. This calibration consists of fitting a straight-line relationship between these two (concentration x-axis and response y-axis) variables [[1], [2], [3], [4], [5], [6]]. A straight-line fit can be achieved either from the conventional statistically less appropriate or even erroneous ordinary least-squares linear regression (OLR) model or from the modern statistically coherent uncertainty-based weighted least-squares linear regression (UWLR) model [6,7].

In the XRF technique, it is mandatory to correct for the matrix effects (absorption, enhancement, and scattering) before determining the final concentrations [[8], [9], [10], [11], [12], [13], [14], [15]]. These effects can be handled through influence coefficients, which can be divided into three basic types [8]: (i) Regression or Empirical model, which has little connection with the XRF theory but allows the determination of influence coefficients by regression analysis of the data sets obtained from reference materials; (ii) Fundamental Parameter model [9], being a totally theoretical approach, requires starting with concentrations and then calculating the intensities; and (iii) Fundamental Algorithm model, based on some simplification of the Fundamental Parameter method, allows concentrations to be calculated from XRF intensities. The weaknesses of the Fundamental Parameter method were pointed out by Rousseau [8], who advocated for the Fundamental Algorithm model [10,11] to be adopted. The Empirical or Regression model were considered less useful [8]. However, the statistically correct weighted least-squares linear regression (WLR) models have not been evaluated in any of the XRF reviews. Therefore, the inference about less usefulness of the regression procedures may not be strictly valid.

In a more recent review, Rousseau [12] evaluated the Lachance–Traill [14] and Claisse–Quintin models [15] as well as the empirical influence coefficients. Rousseau [12] concluded that the empirical method is most suited for use with the Lachance–Traill [14] algorithm and all other proposed algorithms, such as Rasberry and Heinrich [16] or Lucas-Tooth and Pyne [17], etc., are not recommended because of their extremely limited application range or lack of accuracy. These other algorithms, therefore, are not likely to be useful in geochemistry, because the elemental concentrations may vary widely from 0 to 100%, e.g., SiO₂, and, consequently, the other compositional variables will also vary widely [6]. Nevertheless, it may be useful to apply the theoretical Fundamental Parameter method [10,11] under the Claisse–Quintin model [15] and compare it with the empirical model (Lachance–Traill [14] algorithm), along with the OLR and UWLR methodologies. However, for the theoretical model, pure single element materials are required, which are not available to us for the geochemical analysis of major elements or oxides. Rousseau [11] presented a semi-theoretical method, in which the geochemical elements (but not oxides) could be successfully analysed. However, even if successful, the hypothetical example of the major elements presented by Rousseau [11] might not be useful in geochemistry, because oxygen, constituting a large component in all rock and mineral matrices, cannot be easily determined. The presence of oxygen was not even mentioned in the hypothetical examples [11], because all elements, without oxygen, were assumed to sum up to 100%. One may argue that oxygen can be added later to the respective elements as their oxides; however, the conversion of Fe becomes difficult or approximate because of its two oxidation states (FeO and Fe₂O₃) whose proportion is not known a priori. Nevertheless, an option will be to assume all Fe presented as the one or the other oxide. Therefore, the application, evaluation and comparison of a theoretical method with empirical models for the major element determinations in complex rock and mineral matrices should await the availability of all 10 pure elements or components and the theoretical coefficients.

Rousseau [12] explained that the empirical influence coefficients are derived from the best possible fit between the measured intensities and concentrations of a given set of “proper” reference materials. With the increase of the number (m) of analytes, large number of reference materials (at least, 2m + 1) would be necessary. The other basic requirement is that the preparation of the reference materials and the “unknown” samples should be the same and reproducible. There are other basic requirements for the empirical method [12], which can be better fulfilled by the WLR or UWLR than the OLR model [6]. An advantage of the empirical coefficients is that they must be calculated only once, so long as the sample preparation and instrumental conditions are maintained the same. There is no need to resort to complex theory for concentration calculations [12].

In this paper, therefore, we deal with the empirical approach. For the regression models, recently, the two variants (OLR and UWLR) were compared for the calibration of an XRF spectrometer and it was inferred that the UWLR provided more reliable results [18]. It has been argued that the OLR model is statistically inappropriate, because several assumptions are violated [[2], [3], [4], [5], [6], [7]]. The situation is even worse for the XRF method of rock analysis [6]. Unfortunately, the OLR is the only regression model that has been used in the XRF calibrations, except the recent work by [18].

The LaChance–Traill algorithm [14] is the recommended algorithm for empirical coefficients [12]. Although the authors [18] applied both regressions (OLR and UWLR), they used the same matrix effect correction algorithm for both these regressions. To the best of our knowledge, the matrix-correction equations, specifically for the UWLR model [18], have not yet been reported, nor are their implications deduced for the OLR and UWLR models. We also present a comparison of corrections involving 11 α (a conventional number of alpha corrections or empirical coefficients, i.e., a conventional approach in XRF work) and 26 α (a new approach involving large number of alpha corrections) in the matrix solution for the determination of 10 major elements in geological materials. This was facilitated from an unusually large number of calibrators. It is quite possible that such large number (62) of reference materials may have been used earlier (e.g., by Lezzerini et al. [19]) but the newer more reliable (lesser uncertainties, with narrower confidence limits of the mean) concentrations were never derived nor used before in any research, except [18]. Further, the UWLR model [18] was also not used by anyone, including [19].

Recently, Panchuk et al. [20] presented a review on the chemometric methods applied to the XRF data, in which they pointed out the use different regression models, such as partial least-squares regression and several multivariate techniques. All these techniques make use of the data (mean values or most frequently single measurements), without taking in account both the central tendency and dispersion parameters of experimental data. Therefore, all these methods can be grouped under the heading of OLR models [[2], [3], [4], [5], [6], [7]]. The use of WLR [[2], [3], [4], [5]] or UWLR models [6,7] was not even mentioned in this recent review.

Therefore, we compare two different matrix-effect correction algorithms based on the two types of regressions and propose a new UWLR calibration from 62 geochemical reference materials (GRMs) involving 26 α. We also use the same GRMs as unknowns to determine the 10 major element concentrations and their total uncertainties. The quality of these results was thus evaluated in terms of the known concentrations of the GRMs. Besides showing the superiority of the UWLR model, the innovation of our work lies with the possibility of carrying out matrix-effect corrections for this new statistical model and estimating individually the total uncertainty values, along with the mean concentrations for unknown samples, for which 4 additional GRMs, independent of the calibration procedure, were used as unknowns.